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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity Cluster algebras and applications Bernhard Keller

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Cluster algebras and applications

Bernhard Keller

Université Paris Diderot – Paris 7

DMV Jahrestagung Köln, 22. September 2011

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Context

Cluster algebras Fomin-Zelevinsky 2002 [7] Lie theory canonical bases/total positivity [17] [25] Poisson geometry [13] higher Teichmüller th. [5] discrete

  • dyn. syst.[23]

algebraic

  • geom. [20] [28]

repres. theory combinatorics [2] [1] [6] [26]

  • categori−

fication

  • Bernhard Keller

Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Plan

1

Preliminaries: The Dynkin diagrams

2

Definitions: quiver mutation, cluster algebras

3

Application in Lie theory, after B. Leclerc et al.

4

Application to discrete dynamical systems: periodicity

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

The Dynkin diagrams (of type ADE)

Name Graph n

  • Cox. nber

An

  • . . .
  • ≥ 1

n + 1 Dn

  • . . .
  • ≥ 4

2n − 2 E6

  • 6

12 E7

  • 7

18 E8

  • 8

30

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

A quiver is an oriented graph

Definition A quiver Q is an oriented graph: It is given by a set Q0 (the set of vertices) a set Q1 (the set of arrows) two maps

s : Q1 → Q0 (taking an arrow to its source) t : Q1 → Q0 (taking an arrow to its target).

Remark A quiver is a ‘category without composition’.

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

A quiver can have loops, cycles, several components.

Example The quiver A3 : 1

α

2

β

3 is an orientation of the Dynkin

diagram A3 : 1 2 3 . Example Q : 3

λ

  • 5

α

  • 6

1

ν

2

β

  • µ
  • 4

γ

  • We have Q0 = {1, 2, 3, 4, 5, 6}, Q1 = {α, β, . . .}.

α is a loop, (β, γ) is a 2-cycle, (λ, µ, ν) is a 3-cycle.

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Definition of quiver mutation

Let Q be a quiver without loops nor 2-cycles (from now on always assumed). Definition (Fomin-Zelevinsky) Let j ∈ Q0. The mutation µj(Q) is the quiver obtained from Q as follows 1) for each subquiver i

β

j

α

k , add a new arrow

i

[αβ]

k ;

2) reverse all arrows incident with j; 3) remove the arrows in a maximal set of pairwise disjoint 2-cycles (e.g. •

  • yields •
  • , ‘2-reduction’).

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Examples of quiver mutation

A simple example: 1

  • 2
  • 3
  • 1)

1

  • 2
  • 3
  • 2)

1

  • 2

3

  • 3)

1

  • 2

3

  • Bernhard Keller

Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

More complicated examples: Google ‘quiver mutation’!

1 2 3 4 5 6 7 8 9 10.

  • 1

2 3 4 5 6 7 8 9 10

  • 1

2 3 4 5 6 7 8 9 10

  • Recall: We wanted to define cluster algebras!

Bernhard Keller Cluster algebras and applications

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Seeds and their mutations

Definition A seed is a pair (R, u), where a) R is a quiver with n vertices; b) u = {u1, . . . , un} is a free generating set of the field Q(x1, . . . , xn). Example: (1 → 2 → 3, {x1, x2, x3}) = (x1 → x2 → x3). Definition For a vertex j of R, the mutation µj(R, u) is (R′, u′), where a) R′ = µj(R); b) u′ = u \ {uj} ∪ {u′

j}, with u′ j defined by the exchange

relation uju′

j =

  • arrows

i→j

ui +

  • arrows

j→k

uk.

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

An example

x1

x2 x3

1+x2 x1

x2

  • x3

x1

  • x1+x3

x2

  • x3
  • x1

x2

1+x2 x3

  • .

. . . . . . . . . . . . . . . . .

  • µ1
  • µ2
  • µ3
  • µ2
  • µ3
  • µ1
  • µ3
  • µ1
  • µ2
  • Bernhard Keller

Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Clusters, cluster variables and the cluster algebra

Let Q be a quiver with n vertices. Definition a) The initial seed is (Q, x) = (Q, {x1, . . . , xn}). b) A cluster is an n-tuple u appearing in a seed (R, u)

  • btained from (Q, x) by iterated mutation.

c) The cluster variables are the elements of the clusters. d) The cluster algebra AQ is the subalgebra of the field Q(x1, . . . , xn) generated by the cluster variables. e) A cluster monomial is a product of powers of cluster variables which all belong to the same cluster. Remark If Q is mutation equivalent to Q′, then AQ

∼

→ AQ′.

Bernhard Keller Cluster algebras and applications

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Fundamental properties

Let Q be a connected quiver. Theorem (Fomin-Zelevinsky, 2002-03 [7] [8]) a) All cluster variables are Laurent polynomials. b) There is only a finite number of cluster variables iff Q is mutation-equivalent to an orientation ∆ of a Dynkin diagram ∆. Then ∆ is unique and called the cluster type of Q. Examples for a) and b): A3, D4. Positivity conjecture (Fomin-Zelevinsky) All cluster variables are Laurent polynomials with non negative coefficients. Remark Partial results: [16] [29] [4] [27] [3] [30] [24] . . . . Still wide open in the general case.

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¡ ¡

Sergey ¡Fomin ¡ ¡ University ¡of ¡Michigan ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

Andrei ¡Zelevinsky ¡ ¡ Northeastern ¡University ¡

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Construction of a large part of the dual semi-canonical basis

Let g be a simple complex Lie algebra of type ADE and U+

q (g) the positive part of the Drinfeld-Jimbo quantum group.

Theorem (Geiss-Leclerc-Schröer) a) (April 2011 [11]): U+

q (g) admits a canonical structure of

quantum cluster algebra. b) (2006 [12]): All cluster monomials belong to Lusztig’s dual semi-canonical basis of the specialization of U+

q (g) at

q = 1. Remarks 1) This agrees with Fomin–Zelevinsky’s original hopes. 2) Main tool: add. categorification using preproj. algebras.

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

The periodicity conjecture, I

Origin: Alexey Zamolodchikov’s study of the thermodyn. Bethe ansatz (1991). Applications in number theory: identities for the Rogers dilogarithm. Notation ∆ and ∆′ two Dynkin diagrams with vertex sets I, I′, h, h′ their Coxeter numbers, A, A′ their adjacency matrices, Yi,i′,t variables where i ∈ I, i′ ∈ I′, t ∈ Z. Y-system associated with (∆, ∆′) (i, j′) (i, i′) (j, i′) Yi,i′,t−1Yi,i′,t+1 =

  • j∈I(1 + Yj,i′,t)aij
  • j′∈I′(1 + Y −1

i,j′,t)a′

i′j′ . Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

The periodicity conjecture, II

Periodicity conjecture (Al. Zamolodchikov 1991, [31] [21] [22]) All solutions to this system are periodic of period dividing 2(h + h′). Case Authors (An, A1) Frenkel-Szenes (1995) and Gliozzi-Tateo (1996) (∆, A1) Fomin-Zelevinsky (2003) (An, Am) Volkov (2007), Szenes (2006), Henriques (2007) Theorem (K, 2010) The conjecture holds for (∆, ∆′). Tools: Link to cluster algebras, additive categorification using the cluster category constructed from quiver representations.

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Link to cluster algebras: Square product quivers

Example: A4A3

  • µ+ =

sequence of mutations at all ◦. µ− = sequence of mutations at all •. Key Lemma The conjecture holds iff the sequence of seeds . . .

µ− S0 µ+ S1 µ− S2 µ+ . . .

  • f the (principal extension of the) square product Q = ∆∆′ is

periodic of period dividing 2(h + h′).

Bernhard Keller Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Combinatorial periodicity

Theorem Let D be any quiver containing ∆∆′ as a full subquiver. Then (µ+µ−)h+h′ (D) = D. A local and an exotic example local: (A5, A4) : h + h′ = 6 + 5 = 11 exotic: (E6, E7) : h + h′ = 12 + 18 = 30

  • +

+

  • Bernhard Keller

Cluster algebras and applications

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Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity

Summary

Cluster algebras are commutative algebras with a rich combinatorial structure. They have important applications in Lie theory, in discrete dynamical systems and in many other subjects. Do not forget to google ‘quiver mutation’!

Bernhard Keller Cluster algebras and applications

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References

[1] Fr´ ed´ eric Chapoton, Enumerative properties of generalized associahedra, S´ em.

  • Lothar. Combin. 51 (2004/05), Art. B51b, 16 pp. (electronic).

[2] Fr´ ed´ eric Chapoton, Sergey Fomin, and Andrei Zelevinsky, Polytopal realiza- tions of generalized associahedra, Canad. Math. Bull. 45 (2002), no. 4, 537– 566, Dedicated to Robert V. Moody. [3] Philippe Di Francesco and Rinat Kedem, Q-systems as cluster algebras. II. Cartan matrix of finite type and the polynomial property, Lett. Math. Phys. 89 (2009), no. 3, 183–216. [4] G. Dupont, Positivity in coefficient-free rank two cluster algebras, Electron. J.

  • Combin. 16 (2009), no. 1, Research Paper 98, 11.

[5] Vladimir V. Fock and Alexander B. Goncharov, Cluster ensembles, quanti- zation and the dilogarithm, Annales scientifiques de l’ENS 42 (2009), no. 6, 865–930. [6] Sergey Fomin, Michael Shapiro, and Dylan Thurston, Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), no. 1, 83– 146. [7] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J.

  • Amer. Math. Soc. 15 (2002), no. 2, 497–529 (electronic).

[8] , Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. [9] , Y -systems and generalized associahedra, Ann. of Math. (2) 158 (2003),

  • no. 3, 977–1018.

[10] Edward Frenkel and Andr´ as Szenes, Thermodynamic Bethe ansatz and dilog- arithm identities. I, Math. Res. Lett. 2 (1995), no. 6, 677–693. [11] Christof Geiß, Bernard Leclerc, and Jan Schr¨

  • er, Cluster structures on quan-

tum coordinate rings, arXive: 1104.0531 [math.RT]. [12] , Rigid modules over preprojective algebras, Invent. Math. 165 (2006),

  • no. 3, 589–632.

[13] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, vol. 167, American Mathematical Society, Providence, RI, 2010. [14] F. Gliozzi and R. Tateo, Thermodynamic Bethe ansatz and three-fold triangu- lations, Internat. J. Modern Phys. A 11 (1996), no. 22, 4051–4064. [15] Andr´ e Henriques, A periodicity theorem for the octahedron recurrence, J. Al- gebraic Combin. 26 (2007), no. 1, 1–26. [16] David Hernandez and Bernard Leclerc, Cluster algebras and quantum affine algebras, arXiv:0903.1452v1 [math.QA]. [17] Masaki Kashiwara, Bases cristallines, C. R. Acad. Sci. Paris S´

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(1990), no. 6, 277–280. [18] Bernhard Keller, Cluster algebras, quiver representations and triangulated cat- egories, arXive:0807.1960 [math.RT]. [19] , The periodicity conjecture for pairs

  • f

Dynkin diagrams, arXiv:1001.1531. [20] Maxim Kontsevich and Yan Soibelman, Stability structures, Donaldson- Thomas invariants and cluster transformations, arXiv:0811.2435. [21] A. Kuniba and T. Nakanishi, Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), no. 37, 3487–3494. [22] Atsuo Kuniba, Tomoki Nakanishi, and Junji Suzuki, Functional relations in solvable lattice models. I. Functional relations and representation theory, In-

  • ternat. J. Modern Phys. A 9 (1994), no. 30, 5215–5266.

1

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2

[23] Atsuo Kuniba, Tomoki Nakanishi, and Junji Suzuki, T -systems and y -systems in integrable systems, Journal of Physics A: Mathematical and Theoretical 44 (2011), no. 10, 103001. [24] Kyungyong Lee and Ralf Schiffler, A combinatorial formula for rank 2 cluster variables, arXive:1106.0952 [math.CO]. [25] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J.

  • Amer. Math. Soc. 3 (1990), no. 2, 447–498.

[26] Gregg Musiker, A graph theoretic expansion formula for cluster algebras of type Bn and Dn, arXiv:0710.3574v1 [math.CO], to appear in the Annals of Combinatorics. [27] Gregg Musiker, Ralf Schiffler, and Lauren Williams, Positivity for cluster al- gebras from surfaces, arXiv:0906.0748. [28] Kentaro Nagao, Donaldson-Thomas theory and cluster algebras, arX- ive:1002.4884 [math.AG]. [29] Hiraku Nakajima, Quiver varieties and cluster algebras, arXiv:0905.0002v3 [math.QA]. [30] Fan Qin, Quantum cluster variables via Serre polynomials, math.RT/1004.4171. [31] F. Ravanini, A. Valleriani, and R. Tateo, Dynkin TBAs, Internat. J. Modern

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[32] Andr´ as Szenes, Periodicity of Y-systems and flat connections, Lett. Math.

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